The rupture of small blood vessels and other tissue damage during kidney stone treatment with shock wave lithotripsy (SWL) has become a greater concern recently since it was established that newer lithotripters actually create more tissue damage than older lithotripters. Extensive experimentation has shown that the tissue damage is the result of cavitating microbubbles produced within vessels of the kidney in response to the high intensity acoustic pulse used to break up stones. Unrelated to SWL, it was recently shown that significant tissue damage results when ultrasound (US) contrast agents (encapsulated gas microbubbles) are used intravenously with comparatively low power medical ultrasound imaging equipment. In both cases, a lack of understanding of the dynamic behavior of microbubbles in vessels, and their interactions with tissue, has led to unforeseen side effects including vessel rupture and hemorrhaging. In fact, the existing mathematical models that have been used to develop SWL and US contrast agent applications completely neglect the surrounding tissue or vessel and any effects they have on microbubble dynamics. Experiments have recently shown that the dynamic response of microbubbles is indeed substantially affected by the surrounding vessel and elastic tissue. Therefore, the primary objective of the project proposed here is to develop a mathematical model that predicts the dynamics of microbubbles inside vessels and their interactions with tissue. We expect to find that the linear and nonlinear dynamic response of microbubbles to SWL or US stimulation is significantly altered by confinement in a vessel and depends on a variety of parameters including vessel diameter, tissue stiffness, and others. The models developed here will be applied to make accurate, quantitative predictions of microbubble dynamics in a variety of vessel types under SWL and US conditions. The results will be analyzed to identify stresses that may cause damage to vessel walls and other nearby tissue. The developed model will also be applied to predict the effect of vessel confinement on second harmonic imaging with US contrast agents. The long-term goal of this project is to identify procedures and methods to reduce the occurrence of tissue damage and improve the effectiveness of SWL treatment and US contrast agent imaging. Eventually, the models may prove useful in developing new procedures such as targeted drug or gene delivery, tissue etching, and other biomedical applications that rely on the use microbubbles.